Impossible math/geometry problem?

[BaDaBooM]

Ok, I saw this problem (no, not homework.. I'm not in classes right now) and was wondering if anyone else had seen this problem or could solve it. I'll be impressed if someone finds a method/solves it/determinds it's impossible (backed up with how/why):

You have 2 different sized circles in a 2-dimensional plane. The smaller one is inside the larger with both centers at the same point C. The smaller circle can rotate while the larger circle is stationary.

There are 3 different types of points: Type X, Type A, Type B. There must be at least 3 X points at the circumference of the smaller circle, evenly spaced. Type A and B points are put at the circumference of the larger circle as follows. A and B must be placed as a pair, A before B in a clockwise fashion, given a distance Delta apart (not arch length, straight line distance). There must be at least 3 pairs of A and B, but they do not have to be evenly spaced. However, each pair must be at least Delta distance apart.

A CXA radius is defined as when an X point aligns with both point A and C (center) in which a line can be drawn to make a raduis of the larger circle. A CXB radius is defined as when an X point aligns with both point B and C (center) in which a line can be drawn to make a raduis of the larger circle.

The object is to place the X points evenly spaced on the smaller circle and the A-B pairs on the larger circle such that, during a 360 degree clockwise rotation of the smaller circle there are always more CXB radii than CXA radii at any point of the rotation. What is the minimum number of X points, minimum number of A-B pairs and their locations (in degrees), and the radius of the larger circle if:

Delta = .875"?

Delta = 2.25"?

I made a quick (and yes, crappy - I know my artwork sux ) example of a diagram of what they are talking about if anybody was as confused as I was. In my example I have 3 X points 120 degrees apart and 4 A-B pairs... though mine is just an example to help clarify and is no where near a solution. Anybody have any idea how to go about this?

(Sorry about the geocities, quickest way to get it up here)

 

[blahblah99]

well, if you work out the equations, you will end up with a something like a 6x6 matrix in which you will have to solve for solutions.

 

[BaDaBooM]

Originally posted by: blahblah99
well, if you work out the equations, you will end up with a something like a 6x6 matrix in which you will have to solve for solutions.

which equations?

 

[blahblah99]

The equations you are suppose to come up using state-space approach. haha, actually I don't know. I havn't really looked at the problem. Too confusing and its late in the day, so I'm kinda lazy right now.

 

[Fallen Kell]

hmmm...nevermind, I read the question wrong....

Re-reading the question, unless you allow for #CXA = #CXB, this is an impossible question.

But assuming you allow for the above case, you are you asking for #CXA =< #CXB at all points in time. It should be a simple problem once you convert into a circular co-ordinate system (i.e. (theta, r) instread of (x, y)...then the problem turns into a simple state problem with the CXA or CXB event occuring when the co-ordinates of the points have their theta value equal to each other). Things become much easier looking at it that way.